Recall that a cocycle on 𝔞\mathfrak{a} is a closed element in CE(𝔞)CE(\mathfrak{a}). An invariant polynomial is a closed elements in W(𝔞)W(\mathfrak{a}) that sits in the shifted copy ∧•(𝔞*[1])\wedge^\bullet (\mathfrak{a}^*[1]).

This means that for X∈𝔞X \in \mathfrak{a}, for ιX:W(𝔞)→W(𝔞)\iota_X : W(\mathfrak{a}) \to W(\mathfrak{a}) the contraction derivation and adX:=[dW,ιX]ad_X := [d_W, \iota_X] the corresponding Lie derivative, we have in particular that an invariant polynomial ⟨−⟩∈W(𝔞)\langle -\rangle \in W(\mathfrak{a}) is invariant in the sense that

adX⟨−⟩=0.
ad_X \langle -\rangle = 0
\,.

For 𝔞=𝔤\mathfrak{a} = \mathfrak{g} an ordinary Lie algebra, an invariant polynomial on 𝔤\mathfrak{g} is precisely a symmetric multilinear map on 𝔤\mathfrak{g} which is adad-invariant in the ordinary sense.

an invariant polynomial on 𝔞\mathfrak{a} is an elements ⟨−⟩∈W(𝔞)\langle - \rangle \in W(\mathfrak{a}) with the property that

⟨−⟩\langle - \rangle is a wedge product of generators in the shifted copy of 𝔞*\mathfrak{a}^*W(𝔞)W(\mathfrak{a}), i.e.

⟨−⟩∈∧•𝔞*[1]
\langle - \rangle \in \wedge^\bullet \mathfrak{a}^*[1]

or equivalently: for all x∈𝔞x \in \mathfrak{a} and ιX:W(𝔞)→W(𝔞)\iota_X : W(\mathfrak{a}) \to W(\mathfrak{a}) the contraction derivation, we have

ιx⟨−⟩=0;
\iota_x \langle -\rangle = 0
\,;

it is closed in W(𝔞)W(\mathfrak{a}) in that dW(𝔞)⟨−⟩=0d_{W(\mathfrak{a})} \langle - \rangle = 0

or more generally its differential is again in the shifted copy.

Remark

This implies that for

adx:=[dW(𝔞),ιX]
ad_x := [d_{W(\mathfrak{a})}, \iota_X]

the Lie derivative in W(𝔞)W(\mathfrak{a}) along x∈𝔞x \in \mathfrak{a}, which encodes the coadjoint action of 𝔞\mathfrak{a} on W(𝔞)W(\mathfrak{a}), we have

adx⟨−⟩=0
ad_x \langle - \rangle = 0

for all xx. But the condition for an invariant polynomial is stronger than these ad-invariances. For instance there are ∞-Lie algebra cocyclesμ∈CE(𝔤)\mu \in CE(\mathfrak{g}) which when regarded as elements in W(𝔤)W(\mathfrak{g}) are ad-invariant. But being entirely in the un-shifted copy, μ∈∧•𝔤*\mu \in \wedge^\bullet \mathfrak{g}^*, these are not invariant polynomials.

Definition

We say an invariant polynomial is decomposable if it is the wedge product in W(𝔤)W(\mathfrak{g}) of two invariant polynomials of non-vanishing degree.

Definition

Two invariant polynomials P1,P2∈W(𝔤)P_1, P_2 \in W(\mathfrak{g}) are horizontally equivalent if there is ω∈ker(W(𝔤)→CE(𝔤))\omega \in ker(W(\mathfrak{g}) \to CE(\mathfrak{g})) such that

P1=P2+dWω.
P_1 = P_2 + d_W \omega
\,.

Proposition

Proof

Let P=P1∧P2P = P_1 \wedge P_2 be a wedge product of two indecomposable polynomials. Then there exists a Chern-Simons elementcs1∈W(𝔤)cs_1 \in W(\mathfrak{g}) such that dWcs1=P1d_W cs_1 = P_1. By the assumption that P2P_2 is in non-vanishing degree and hence in ker(W(𝔤)→CE(𝔤))ker(W(\mathfrak{g}) \to CE(\mathfrak{g})) it follows that

also cs1∧P2cs_1 \wedge P_2 is in ker(W(𝔤)→CE(𝔤))ker(W(\mathfrak{g}) \to CE(\mathfrak{g}))

Definition

Since invariant polynomials are closed, the inclusion of graded vector spaces from observation 2 induces an inclusion (monomorphism) of dg-algebras

inv(g)↪W(g).
inv(g) \hookrightarrow W(g)
\,.

Examples

On Lie algebras

Observation

For 𝔤\mathfrak{g} a Lie algebra, this definition of invariant polynomials is equivalent to more traditional ones.

Proof

To see this explicitly, let {ta}\{t^a\} be a basis of 𝔤*\mathfrak{g}^* and {ra}\{r^a\} the corresponding basis of 𝔤*[1]\mathfrak{g}^*[1]. Write {Cabc}\{C^a{}_{b c}\} for the structure constants of the Lie bracket in this basis.

Then for P=P(a1,⋯,ak)ra1∧⋯∧rak∈∧r𝔤*[1]P = P_{(a_1 , \cdots , a_k)} r^{a_1} \wedge \cdots \wedge r^{a_k} \in \wedge^{r} \mathfrak{g}^*[1] an element in the shifted generators, the condition that it is dW(𝔤)d_{W(\mathfrak{g})}-closed is equivalent to

on the Lie algebra 𝔤\mathfrak{g} canonically identifies also with an invariant polynomial of the string Lie 2-algebra. But the differnce is that the Killing form⟨−,−⟩:=Pabra∧rb\langle -,- \rangle := P_{a b} r^a \wedge r^b is non-trivial as a polynomial on 𝔤\mathfrak{g}, but as a polynomial on 𝔤μ3\mathfrak{g}_{\mu_3} becomes horizontally equivalent ,def. 3), to the trivial invariant polynomial.

Proposition

Proof

Let cs3∈W(𝔤)cs_3 \in W(\mathfrak{g}) be any Chern-Simons element for ⟨−,−⟩\langle -,- \rangle, hence an element such that

cs3|CE(𝔤)=μ3cs_3|_{CE(\mathfrak{g})} = \mu_3;

dWcs3=⟨−,−⟩d_W cs_3 = \langle -,- \rangle.

Then notice that by the above we have in W(𝔤μ3)W(\mathfrak{g}_{\mu_3}) that the differential of the new generator hh is equal to that of μ3\mu_3:

dWh=dWμ3.
d_W h = d_W \mu_3
\,.

We on 𝔤μ4\mathfrak{g}_{\mu_4} we can replace μ3\mu_3 by hh and still get a Chern-Simons element for the Killing form:

cs˜3:=cs3−μ3+h.
\tilde cs_3 := cs_3 - \mu_3 + h
\,.

dWcs˜3=⟨−,−⟩.
d_W \tilde cs_3 = \langle -,- \rangle
\,.

But while μ3\mu_3 is not in ker(W(𝔤μ3)→CE(𝔤μ3))ker(W(\mathfrak{g}_{\mu_3}) \to CE(\mathfrak{g}_{\mu_3})), the element hh is, by definition. Therefore cs˜3\tilde cs_3 is in that kernel, and hence exhibits a horizontal equivalence between ⟨−,−⟩\langle -,- \rangle and 00.

where {ta}\{t^a\} is a dual basis in degree 1 for some semisimple Lie algebra𝔤\mathfrak{g} as above, bb and cc are generators in degree 2 and 3, respectively, and μ3∝⟨−,[−,−]⟩\mu_3 \propto \langle -,[-,-]\rangle is the canonical Lie algebra cocycle in degree 3, as above.

So the Lie 3-algebra (bℝ→𝔰𝔱𝔯𝔦𝔫𝔤)(b \mathbb{R} \to \mathfrak{string}) is a kind of resolution of the ordinary Lie algebra 𝔤\mathfrak{g}. It is for instance of use in the presentation of twisted differential string structures, where the shifted piece bℝb \mathbb{R} in (bℝ→𝔰𝔱𝔯𝔦𝔫𝔤)(b \mathbb{R} \to \mathfrak{string}) picks up the failure of 𝔰𝔬\mathfrak{so}-valued connections to lift to 𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{string}-2-connections.

The proof of the following proposition may be instructive for seeing how the definition of horizontal equivalence of invariant polynomials takes care of having the invariant polynomials of (bℝ→𝔰𝔱𝔯𝔦𝔫𝔤)(b\mathbb{R} \to \mathfrak{string}) agree with those of 𝔤\mathfrak{g}.

(if another normalization is chosen, then the corresponding factor will float around the following formulas without changing anything of the end result).

Now the indecomposable invariant polynomials are those of 𝔤\mathfrak{g} and one additional one: gg. This means that before deviding out horizontal equivalence on generators, the invariant polynomials of (bℝ→𝔰𝔱𝔯𝔦𝔫𝔤)(b \mathbb{R} \to \mathfrak{string}) are not equal to those of 𝔤\mathfrak{g}, due to the superfluous generator gg.

Notice that the homotopy cs−μ3+hcs - \mu_3 + h here is indeed in ker(W(𝔤)→CE(𝔤))ker(W(\mathfrak{g}) \to CE(\mathfrak{g})): the component of cscs not in that kernel is precisely μ\mu. The above formula subtracts this offending summand and replaces it with the new generator hh, which by definition is in the kernel and whose image under dWd_W is the image of μ\mu under dWd_W, plus the superfluous new generator of invariant polynomials.

Therefore in horizontal equivalence classes of invariant polynomials on (bℝ→𝔰𝔱𝔯𝔦𝔫𝔤)(b \mathbb{R} \to \mathfrak{string}) the superfluous gg is identified with the Killing form ⟨−,−⟩\langle-,- \rangle, and hence the claim follows.

Properties

As differential forms on the moduli stack of connections

The invariant polynomials of a Lie algebra 𝔤\mathfrak{g}, thought of as equipped with trivial differential, are the de Rham complex of differential forms on the universal moduli stack BGconn\mathbf{B}G_{conn} of GG-principal connectionsFreed-Hopkins 13.

This graded vector space has a vector space isomorphism of degree -1 to the graded vector space of add generators of the Lie algebra cohomologyH•(mthafrakg)=H•(CE(𝔤))H^\bullet(\mthafrak{g}) = H^\bullet(CE(\mathfrak{g})).

Transgression cocycles and Chern-Simons elements

Definition

(Chern-Simons elements and transgression cocycles)

Let 𝔞=𝔤\mathfrak{a} = \mathfrak{g} be an ∞-Lie algebra. Since the cochain cohomology of the Weil algebraW(𝔤)W(\mathfrak{g}) is trivial, for every invariant polynomial ⟨−⟩∈W(𝔤)\langle -\rangle \in W(\mathfrak{g}) there is necessarily an element cs∈W(𝔤)cs \in W(\mathfrak{g}) with

where bn−1ℝb^{n-1}\mathbb{R} denotes the ∞-Lie algebra whose CE-algebra has a single generator in degree nn and vanishing differential, and where CE(bnℝ)=inv(bn−1ℝ)CE(b^n \mathbb{R}) = inv(b^{n-1}\mathbb{R}) is the algebra of invariant polynomials of bn−1ℝb^{n-1} \mathbb{R}.

Proposition

The element μ∈CE(𝔤)\mu \in CE(\mathfrak{g}) associated to an invariant polynomial ⟨−⟩\langle -\rangle by the above procedure is indeed a cocycle, and its cohomology class is independent of the choice of the element cscs involved.

Proof

The procedure that assigns μ\mu to ⟨−⟩\langle- \rangle is illustarted by the following diagram

We say that μ∈CE(𝔞)\mu \in CE(\mathfrak{a}) is in transgression with ω∈inv(𝔞)⊂CE(Σ𝔞)\omega \in inv(\mathfrak{a}) \subset CE(\Sigma \mathfrak{a}) if their classes map to each other under the connecting homomorphismδ\delta:

δ:[μ]↦[ω].
\delta : [\mu] \mapsto [\omega]
\,.

Example. In the case where 𝔤\mathfrak{g} is an ordinary semisimple Lie algebra, this reduces to the ordinary study of ordinary Chern-Simons 3-forms associated with 𝔤\mathfrak{g}-valued 1-forms. This is described in the section On semisimple Lie algebras.

The diagrams on the left encode those 𝔤\mathfrak{g}-valued forms on U×ΔkU \times \Delta^k whose curvature vanishes on Δk\Delta^k. One can show that one can always find a genuine∞\infty-connection: one for which the curvatures have no leg along Δk\Delta^k, in that they land in Ω•(U)⊗C∞(Δk)\Omega^\bullet(U) \otimes C^\infty(\Delta^k). For those the above diagram extends to